Quantum Monte Carlo for “Difficult” Systems in Materials

Chemistry

Ainsley A. GibsonHoward UniversityWashington, DC 20059

Black Box Computing

• Pros:– Widespread adoption of techniques– Relative ease of use– Always gets a number as output

• Cons:– Often promotes misconceptions– Usually no error estimation– Always gets a number as output

State-of-the-Art Computing

• Pros:– End results are well-analyzed– Results are frequently great!– Near-complete explanation

• Cons:– Expensive (human, not CPU) cost– Not for everyone– Potentially highly selective

“Golden Box” Computing

• Lies somewhere between black box and state of the art.

• Use of high level techniques in a generalized form.

• Tradeoff between high accuracy/high expertise and variable accuracy/low expertise.

MethodElectron Correlation,

in Principle

Electron Correlation,

in Practice

Density Functional Theory

Characteristic density and exact density functional

recover system’s properties

Exact functional unknown, functionals generated by fit

to experiment or theory

Traditional ab initio post-HF methods

Infinite excitations from reference state(s) provide approximation from one-

electron basis

Truncated number of excitation types; selected reference state(s) used

Quantum Monte Carlo (QMC)

Random sampling of wavefunction-based probability in real 3-dimensional space

Explicit inter-particle interaction added to

independent-particle trial functions

Method Pros Cons

Density Functional Theory

Inexpensive, and functionals exist that are

well-tuned to specific chemistries

No hierarchy of functionals, low to medium accuracy

Traditional ab initio post-HF methods

High to very high accuracy; has a hierarchy of methods

Moderately expensive to extremely expensive, may

fail regardless

Quantum Monte Carlo (QMC)

Massively parallel, very high accuracy, simple error

estimation, and simple excited state energies

Expensive to very expensive, small energy differences challenging

This Work…

• There is a significant degree of “art” in QMC calculations, due to the lack of strict restriction on trial function form.

• We wish to determine the degree of necessary “art” in trial function form.

• We also wish to retain the ability to accurately describe “difficult” systems.

Difficult? This isn’t rocket science…

• Typical “difficult” systems have:– Ionized or excited states– Radical or metallic character– Significant delocalization or resonance

• More broadly, “difficult” systems require use of an atypical variant of the technique that need not be used for 95% of chemical systems.

Applications

• Atomic Excited States

• Beryllium Dimer

• Nanoscale Ternary Compounds (HU-CREST)

• Transition Metal Energetics (AHPCRC)

• Atmospherically Interesting

Atomic Excited States

• Simple test of ability to describe electronic structure

• Some reactions require accurate description of excited states

• “Proof of capability” study for future applications to molecular systems

Atomic Excited StatesDeviation from experiment, eV

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Li 2P Be 3P Be 1P B 4P C 1D C 1S N 2D N 2P O 1D O 1S F 4P

B3LYP MP2 CCSD CCSD(T) DMC

Beryllium Dimer

• Poorly described by simpler traditional basis set ab initio techniques.

• Multi-reference character due to 2s-2p near degeneracy.

• Motivated by prior success with atomic excited states.

• A few-electron system amenable to all-electron fixed-node DMC.

Dissociation Energies, in cm-1

Method De, cm-1

MRSDCI1 1049

R12MRCI, MRACPF2 898(8)

estimated FCI/cc-pV5Z3 803.61 - 822.71

“extensive” ab initio3 944(25)

VMC/CASSCF(4,8) 90% -867(71)

VMC/CASSCF(4,8) 95% -7422(152)

DMC/CASSCF(4,8) 90% 1293(52)

DMC/CASSCF(4,8) 95% 829(64)

Experiment4 839(10)

Nanoscale Ternary Compounds

• Formation of novel compounds at the nanoscale have been proposed.

• The reactions use carbon and oxygen in the presence of a nitrogen plasma.

• We propose to predict some basic properties of proposed reactions and compounds using QMC techniques.

Higher Excited States

• Reactions proposed may proceed through excited and/or ionized states.

• QMC offers the allure of unprecedented accuracy for ionized and excited species.

• QMC is generalized for any electronic state.

• The higher states of nitrogen are first in a series of excited state calculations.

Nitrogen Excited States

CISD/cc-pVTZ

VMC DMC Exp’t

2D 2.9227 2.50(3) 2.43(4) 2.3835

2P 3.1458 3.34(4) 3.45(4) 3.5756

4P (2s2p4) 10.6570 10.95(6) 10.84(4) 10.9239

Transition Metals

• When carefully chosen, there are methods able to describe selected metallic systems.

• Satisfaction with price, performance and general applicability is elusive.

• QMC shows promise for metallic systems, and has three features in its favor:– System-independent methodology– Consistent error estimates– Ideal for HPC environments

X-alpha, LSDA, and PL density functionals

IP Deviation, X-alpha, LSDA, PL

-5.00

-4.00

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

Sc Ti V Cr Mn Fe Co Ni Cu Zn

X-alpha LSDA PL

B- functionals: B971, BLYP and BPW91

IP Deviation, B Functionals, eV

-4.00

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

Sc Ti V Cr Mn Fe Co Ni Cu Zn

B971 BLYP BPW91

B1- functionals: B1B95 and B1LYP

IP Deviation, B1 Functionals

-3.50

-3.00

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

Sc Ti V Cr Mn Fe Co Ni Cu Zn

B1B95 B1LYP

B3- functionals: B3P86, B3PW91, B3LYP

IP Deviation of B3 Functionals, eV

-4.00

-3.50

-3.00

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

Sc Ti V Cr Mn Fe Co Ni Cu Zn

B3P86 B3PW91 B3LYP

Selected post-HF Ionization Potentials

IP Deviation, eV

-6.0

-4.0

-2.0

0.0

2.0

Sc Ti V Cr Mn Fe Co Ni Cu Zn

ROHF MP2 CCSD CCSD(T)

QMC/HF Ionization Potentials

IP Deviation, QMC/HF Orbitals

-6.0000

-4.0000

-2.0000

0.0000

2.0000

4.0000

6.0000

8.0000

Sc Ti V Cr Mn Fe Co Ni Cu

Devia

tion

, eV

VMC DMC

2.67,VMC

1.52,DMC

QMC/NO Ionization Potentials

IP Deviation, QMC/Natural Orbitals

-3.5000

-2.5000

-1.5000

-0.5000

0.5000

1.5000

2.5000

3.5000

Sc Ti V Cr Mn Fe Co Ni Cu

VMC DMC

1.71,VMC

1.57,DMC

Ozone Dissociation Energy

• Traditional ab initio has difficulties:– Resonance character of ozone– Low-lying excited state contributions

• Estimates of the dissociation limit are relatively small (1.02 – 1.13 eV).

• Various excited states lie above and below the dissociation limit.

Results to Date, in eV

MRCI 0.943

MRCI+Q 1.049

VMC 0.70(4)

DMC 1.06(16)

Exp. 1.0625(4)

Exp. 1.132(1)

HU-CREST Current Work: Novel Nanoscale Compounds• Characterization of excited states:

– CN, CO, NO, N2, C2, O2

– ONC, OCN

• Energetic profile of proposed reactions

• Large-scale network compounds

AHPCRC Current Work:Transition Metals

• Electron Affinity• Proton Affinity

• Small Clusters, Mx, x = 2,…,10

• Surfaces and solids• Silver nanoparticle stability (collaborative with

CREST)

Future/Current Work:Atmospherically Interesting

• Ozone dissociation and excited state characterization

• S4 inter-conversion energetics

• Excited and ionized states of binary (O, N, C) compounds as atmospheric species

Acknowlegements

• John A. W. Harkless*• William Lester, Jr.• James Mitchell• William Hercules• Floyd Fayton • Gordon Taylor • José González• Mike Towler

• NSF CREST Center for Nanomaterials Characterization and Design

• Army High Performance Computing Research Center

• Computer Learning and Design Center

• TTI